What is the ruling on a father's secret marriage, what are the duties of children towards their father and his wife, and is he entitled to impose his wife on the children to live with them, given the family's rejection of her?

If the father's second marriage fulfills its legitimate conditions, then it is valid. The father has the right to polygyny as long as he does not fear injustice between them, as stated in the Quran. What the father did does not diminish his right to the children's dutifulness and kindness towards him, and the family does not have the right to reject his action. As for housing the second wife with the first in the same house, the father does not have this right except with the mother's consent. Each wife must have an independent section with its facilities in the house. Otherwise, it is not permissible to house them together in one dwelling; rather, he must provide each with an independent dwelling.